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Full text of "Calculus"

A 303 

113 

917 

nsuer 

book 

opy 1 



303 

ANSWER BOOK 

17 TO ACCOMPANY 

suer MARCH AND WOLFF'S CALCULUS 

°° k BY VyJ^ 

py 1 HERMAN W^ilARCH 

and y 

HENRY C>"WOLFF 



Copyright, 1917, by the McGraw-Hill Book Company, Inc. 



Page 11, §12 

1. (a) y =2(x - 2)2; y = 2(x + 3) 2 ; y = 2x 2 + 5; y - 2x 2 - 1; 

y = 2(x + 2) 2 - 1. 

(b) y = - 3(s - 2) 2 ; y = - 3(x + 3) 2 ; y = - 3a; 2 + 5; 

y = -3x 2 - l;y = -3(x + 2) 2 - 1. 

(c) ?/ = log (x - 2); y = log (x + 3); y = log a; + 5; 

2/ = log x - 1; y = log (x + 2) - 1. 

(d) y = e~ x+2 ; i/ = e" 35 " 3 ; y = e~* + 5; y = e~ x - 1; 

2/ = e~ x ~ 2 - 1. 

2. (a) y = 2x 2 + §x; 2/ = 2x 2 - £x; y = 2x 2 + x; y = 2x 2 - x. 

(b) y = - 3x 2 + \x\ y = - 3x 2 - \x\ y = - 3x 2 + 3; 

?/ = - 3x 2 _ x> 

(c) ?/ = log x + \x\ y = log x - \x) y = log x + x; 

2/ = log £ — x. 

(d) y = e~ x + \x\ y = e~ x - Jo:; 1/ = e~ x + x; y = e~ x - x. 

3. (0) y - - 2x 2 ; y = 2x 2 ; x = 2?/ 2 ; x = - 2y 2 . 

(b) y = 3x 2 ; y = - 3x 2 ; x = - 3?/ 2 ; x = 3?/ 2 . 

(c) ?/ = log - ; y = log ( - x) ; x = log 2/; x = - log ( - y). 

(d) y = — e~ x ; 1/ = e z ; x = e~ y ) x = — e y . 

4. (a) p = a sin ( — ~) ; p = a sin f — ^j ; p = — a cos 0; 

p = — a sin 0; p = a cos 0. 

(6) p = a cos ( — o) ; p = a cos (0 — A ; p = a sin 0; 

p = — a cos 0; p = — a sin 0. 
6. (a) 2x7/ = a 2 ; (6) 2x?/ = - a 2 ; (c) y 2 - x 2 = a 2 ; 

(d) y 2 - x 2 = a 2 ; (e) x 2 + y 2 = a. 

Page 26, §20 

j 1 . - * 

" 2\/x - 2 



/ft 



f 



RESULTS OF EXERCISES Yx^ 

Page 39, §30 ^\\*V 9^ 



1. 0.27 of 1 square inch per second. ^ O* 5 ^ 

2. 1.56 square inches per minute. O^* 

3. 0.0051 of 1 inch per minute; 0.8 of 1 square inch per minute. 







Page 42, §32 


12. 14a; (5 - 2x 2 )~% 




15. -±±x(x 2 + 1)' 


17. - 8x(x 2 + 4)" 3 . 




18. 18o; 2 (5 - a; 3 )" 3 . 
Page 46, §36 


3. * 4. 


X 

y 


b 2 x 

5. — —k -' 

a 2 y 



'• - (?)'• 



Page 51, §37 



v& 



39. - i(4 - 3a; 2 ) 2 + C. 41. - A(4 - 3a;) 2 + C. 

42. f(o; 3 + 3a; - 7)* + C. 45. J (6 + 4z - 10a; 2 )* + C. 

48. - J(2 - a; 2 ) 4 + C. 49. 3y = a; 3 + 1. 

50. 32/ = 2a;* + 4(3 - a/2). 51. xy = 1. 

Page 55, §38 

1. 1610 feet. 2. 1710 feet; 332 feet per second. 

3. 257.6 feet down; 966 feet below the starting point; 1094.8 feet. 

4. (1°) v = Z 2 ; s = \t\ 

(2°) v = t 2 - 2; s = %t* - 2t + 3. 

Page 58, §40 

^ 2x 2 + x - 2 12 + 4a; - 5a; 2 

9. 7=-- 10. -5— • 

V a; 2 - 2 3(3.2 _ 4) s 

8 - 3a; - 4a; 2 1 - 2a; 2 

11. 7 — 12. 



\/4 - a; 2 Vl 



a; 



13 dy = 3(l + ^) _ 14 ^ = y {2x " %) . 

'da; 7 — 3a; 3 ?/ 2 * " da; a;(6?/ — x) ' 

di _ 2yVx + yVy 16 ^ _ y(2x - 1) 

^ 2*V|j + * V* " dx x{l ~ x) 

Page 59, §41 
-2 (z-3)(s-l) . 1 + 2a; - x 2 



{x-l) 2 ' (x-2) 2 . "" (1-a;) 2 

JUL -2l9l7©aA467695. 



3- x 



RESULTS OF EXERCISES 

3x 2 + i 



7. 



2(1 +x)Wx- 1 
x + \/x 2 - 1 



-sA 



- 12s 

(x 2 +l) 2 ' 



5. 



8. 



2x* 



4x — 4 



(a; -2)2 
Page 60, §42 

8. - 7x(x 2 + 1)~- 
Page 62, §63 



6. 



9. 



2 + x 




2(1- x) f 




2(x 3 - 3x 2 - 


-4) 



(x - 2)2 



9. x(l - x 2 ) 



Ex. 



Increasing when 



Decreasing 
when 



Maximum 
value 



Minimum 
value 



1. 


x > 
x < and x > 


x < 







2. 






3. 


3 < 


s > 







4. 


For all values of x 








5. 


x < and x > f 


< a? < | 


- 6 




6. 


x > J 


x < § 




2 5 


7. 


x < 1 and a: > 2 


1 < x < 2 


- 5 


- 6 


8. 


s < 1 and s > 1 


- 1 <3 < 1 


9 


5 


9. 


x < — 1 and a > ^ 


- 1 < X < % 





— #1 


10. 


3 < 


x > 







11. 

13. 
14. 



17. 

21. 

24. 

40. 
47. 



4 inches. 12. 2.4 feet by 5 feet. 

When t — 4; Approaching at the rate of 5 miles per hour; Sepa- 
rating at the rate of 7.905 miles per hour. 
6 inches by 6 V3 inches. 

Page 63, §63 

(x - 1) (3x - 1). 18. s(2 - 3x). 19. (1 - x 2 ) 2 (1 - 7x 2 ). 

20. 2(1 - x 3 ) (x - 2) (1 + 6x 2 - 4x 3 ). 



l + 2x 



(x 2 + l) 2 
a 2 

(a 2 - x 2 )^' 



22. - ,. , „. x . x 23. i(x - 1) 3 . 



25. 



(1 + x 2 ) 2 
1- 2x 2 
Vl - x 2 ' 



39. Kl -sT'+C. 



- \{2x - x 2 ) + C. 43. - 2vT- x + C. 



46. 



9x 
16y 



48. f V3 feet per second. 



RESULTS OF EXERCISES 



49. 0; 0.201; 2.27; oo . 50. 18.08. 

51. x = 14.14; angle = 45°. 52. - gt cot 20° = - 88.4*. 

53. 0; - 494.3; - 695. 54. 18.32. 

55. 12.53; 13.96; A of one hour. 56. 9; 27.58; when t = 1.484. 

57. 38.43. 58. 500. 59. 249.5. 60. - 840. 

61. 2tt; 54; 2.16. 



1. (a) 

2. (a) 



Su 



VV + 7 

y 



2(y* - 3)* 

32/ 
7y — 4z 3 
4?/ 3 - 7x 





62. 11.33; 15.1 


2.86; 


153.7. 




Page 67, §45 








b) 


2(6u 2 '+ 5) (x - 


-1). 


(c) - 


2xu 
(u* - 5)* 




»-! 


1/2+2)3 

4y 








(d) 1. 










(b) - 


3?/ 3 + lSx 2 y + Sx 
9xy 2 + 6z 3 




Page 68, §46 








2. 


- /2-n. 

n l 
Page 69, §47 


3. - 


- iu~K 





(c) ■ 
3. (a) 

1. i. 



1. 2V2; 2V2; 2; 2. 

2. y = x + 1, equation of tangent line; 

y = — a; + 3, equation of normal line. 

3. 9x+ 20y = 75. 4. y* = 8x + 1. 
5. V5. 6. iVl7. 7. 2/ 2 



IV17. 
Page 74, §49 



2x +7. 



Ex. 


Maximum 
point 


Minimum 
point 


Point of inflection 


1. 

2. 
3. 


(0,0) 

(-2,4) 


(2, - 4) 
(0,0) 


(1, - 2) 
(-1,2) 
( - t, 5) 


4. 




(1, - 2) 


(0, 1) and (f, - *f ) 

(0,0) 

(_ i _ 13 1) an d (| 18 J) 


5. 




6. 




(-1, -35f) 

\2l 87 


7. 




(0, 1) and (i If) 







RESULTS OF EXERCISES 

Page 77, §50 
1. fm; ¥; 10J. 2. }; |a 2 ; J; T V 3. 2.797. 

Page 81, §51 

1. 20 foot-pounds. 2. 1136 foot-pounds. 3. 119 ergs. 

4. 8920 foot-pounds. 5. 20.83 foot-pounds. 

Page 87, §54 
1. 45°. 

Page 99, §62 

5. M* 2 - 2)~hx. 6. 7^-~ dx - 

2y/2{x - 1)3 

7. (x + 1) O - l) 2 (ox + l)dz. 8. S(x 2 + x - 2) 2 (2x + l)dx. 

9. - i(x - l)~-dx. 10. - Sx(x 2 - l)~ f dr. 

19. y = Cx. 20. ?/ = CSA 2 - 1. 

21. i/ = C(x 3 - 3z 2 + 12x - 2)3. 

Page 102, §63 

z 2cte 



3. Vl + 4a: 2 dx. 4. -^ 1 + JL 



efo. 
4x 



T.^/l+^dx. 8. \jl+^dx. 

Page 105, §65 
1. 14. 2. f. 3. ia 3 (2\/2 - 1). 

Page 109, §67 

2. 121,100 foot-pounds. 3. 21,880 foot-pounds. 

4. 160 inch-pounds. 5. 1012.5 inch-pounds. 
6. 19,475m + 3950 foot-pounds. 

Page 110, §68 

1. 27tt. 2. *irr\ 3. 4Stt; 64tt. 4. 64tt. 

5. ftWo». 6. Va 3 . 7. }tt. 



RESULTS OF EXERCISES 
Page 112, §69 



1. 


1.736. 


2. 


6a. 3. a 


J Va 2 - x 2 


4. 


16.67. 




Page 113, §70 




1. 


16V6tt. 


2. < 


txa 2 . 3. ^tt. 
Page 116, §72 


4. -\ 2 -ira 2 . 


1. 


6000; 18,500 pounds. 


2. 5208 pounds. 










Page 121, §74 




1. 


JV3. 2. 


1.724. 


3. 6. 4. 30. 


5. i 


6. 


17.9. 8. 


5k. 


9. 2g. 10. 100. 


11. 6|; 7§ 


L2. 


-r , where r is the radius. 





Page 130, §76 

^~ ~ . ~ , n ^ . „ x -, - ~ 2 cos 2 2x-\- 4 sin 2 2x , 

12. 2 sin 2z(sec 2 2x + l)dx. 13. f^ dx. 

K J cos 3 2x 

14. (2s + 3) cos (x 2 + dx - 2)dx. 

15. 2(3 - x) sin (3 - x) 2 dx. 

cos 2s 

16. / . ~~~ dx. 17. (cos 2x — 2x sin 2x — 2 sec 2 2x)dx. 
V sin 2x 

20. cot* (2a? - 1) [cos 2 (2s - 1) - 3 sin 2 (2x - l)]dx. 

31. 2. 32. Tra 2 . 33. 8a. 

34. 37ra 2 . 35. Tra 2 . 36. irab. 

37. - tan 0; 6a. 38. Jtt 2 . 

39. 4.37 feet per second. 40. 5.364 feet per minute. 

41. 2 ^Z'K 42. - 4?-. 

Sy-2 cos y 

44. 3a* cos 30 + ,,' sin . _ c _ 

2p cos 1 6 L J 

46. ^[sin 7x + 7 sin z] + C. 

47. Jy[7 cos 3x - 3 cos 7x] + C. 

48. T io[H sin 5z - 5 sin lis] + C. 

49. ^[11 cos 3z - 3 cos lis] + C. 

50. 5 i 6 [2 sin Ux + 7 sin 4s] + C. 

51. 7^^-^ £T [(co — a) cos (a> + a)£ + (a> + a) cos (w — a)t] + C. 



RESULTS OF EXERCISES 
1 



-^ [(co — a) sin (co + a)t + (co + a) sin (co — a)* 1 ] + C. 
53. 



"• 2(co 2 - a 2 ) 
2 









Page 134, §77 


7. 


- 1 
2a: 2 - 2x + 1 




2(*-l) 


' Vi - (i - x) 4 


9. 


2x 




X 


(x 2 - 3) V* 4 - 


6x 2 


10. / 4- sin * x. 

+ 8 Vl - z 2 


11. 


itan-i| +C. 




15. § tan" 1 x 2 + C. 


16. 


fsin" 1 2x +C. 




2r 
17. | sin" 1 -3- + C. 


19. 


| sin" 1 a: 2 + C. 




AFT ^ 

37 '6- 


38. 


2wa 2 . 




39. 0.848. 
Page 139, §79 



2. 250tt feet per second. 

3. — IOOtt radians per minute per minute; 6 J revolutions. 

4. 200tt revolutions. 

v 2 

5. rco 2 radians per second per second, or — feet per second per 

second, where v is the tangential velocity; r is measured as feet and 
time as seconds. The acceleration is directed toward the center of 
the wheel. 

6. rco feet per second, if r is measured as feet and co as radians per 
second. The point is moving in a direction perpendicular to the 
line joining it and the point of contact of the wheel with the hori- 
zontal track. 

7. (x - b) 2 + (y - c) 2 = a 2 . 





Page 144, §81 


1. (a) |t; 2 + *y 


= 0. (b) Itt; d2 d y t2 + 9y = 0. 


2. (a) y = A: sin (3* 


- a). (b) y = k sin (\/3£ — a). 




Page 148, §82 


7 /,Z^n«o ~ c\ \ 


iA 2x(7+:r 2 ) 



1 - X 4 



8 RESULTS OF EXERCISES 

14. e~ x (cos x — sin x). 18. £ 4 5 x (5 + % log 5). 

34. log (e x + e~ x ) + C. 35. 10 log 2 = 6.93. 



37. 


y = Ce*. 




38. 0.4343. 


39. 


— log (esc x + cot a: 


) +'C. 


46. 0.732. 


49. 


(0.4343) (10)* + C. 




51. log| = 0.451. 


55. 


3(a 2 +z 3 ) ^ U * 




63. log 2 = 0.693. 






Page 


154, §83 




a;- 119 




(3 + 3) (133- Sx- 5x 2 ) 


1. 


35(z + 1) (a; - 5)* 


(x - 4)\x - 5) 5 


Q 


17x + 29 




A 2 + x - 5x* 
4. 



6(z + l)'(2a; + 5) 4 2 ^ 1 ~ x 

5. x n ~ l n x {n + £ log n). 6. x sin x 1- cos x log a; • 

7. 10 3 '- 2 [3(7£ + 3) log 10 +7]. 

8. \x S//x ~ h {2 + logz). 9. 7. 10. -• 



11. 


12. -• 13. -• 14. k. 15. h log 1( 

XXX 


16. 


\ du \ dv 1 du 1 dv 1 dw 
u dx v dx u dx v dx w dx 


19. 


y = Cx\ 20. y = Cx\ 21. Ce fF{x)dX . 22. 2/ = Ce* 




Page 158, §84 


1. 


. 1, &2 

g<r*«. 3. ^log^jno. 


4. 


-oe-K 5. , "° • 



Page 159, §85 

1. Joe" - 02 '. 

2. 8.51, assuming that the medium surrounding the body is at zero 
degrees temperature. 

Page 165, §89 

1. f. 2. 2i 3. ft. 4. 16.1; 1. 



1. — tan -1 m; 



RESULTS OF EXERCISES 

Page 168, §90 

Wjsin ft — m cos ft) 

Page 169, §92 

1. Maximum value = 2; Minimum value = — 9. 

2. A square of 200 units. 

Page 175, §96 

1. 7.92 inches. 2. 2\ miles from A. 3. 45°. 

4. 19.73 feet. 5. 11.23 feet, 6. 24 feet. 

9. -• 10. 66° 3.6'. 

7T 

11. 6\/3 inches by 6 inches. 12. 4\/6 inches by 4\/3 inches. 
14. Part 8, |. 

Page 181, §97 

1. - 6. 2. B. 3. -• 4. * = I 

a 2 





5. * = 7T - 2 « 


6. * = ^ + e - 
Page 182, §98 


a' 




1. 2;ra. 


2. 8a. 3. | 


\wa. 




^^^-l,; 


a l J 






Page 183, §99 






1. Tra 2 . ' 2. 67ra 2 . 


3. 67ra 2 . 4. 2 2 5 7T. 




5. * IT. 


6. ,Wa 2 . 7. 5. 


ft 4 3.2 Q 10r > 

8. 37r 3 a 2 . 9. — i— 

IT 4 

Page 187, §100 




10. fc» 


28. p - \t* + t + 2 log (« + !) + C. 







42. yg5 log V7 + Vto + c 
70 - N : - v/5* ' 
69. i l..« Bee. (2*« - 5) + C. 71. J tan" 1 (i sin x) + C. 

73. a*X - EaV + J<W - \x* + C. 



10 RESULTS OF EXERCISES 

84. | 87. log (a; 2 + 9) + tan" 1 | + C. 

88. ^ 2 + 3a; - f log (x 2 + 9) - V tan" 1 ! + C. 
102. x - 2 log (2a; + 7) + C. 
117. § log (rf - 9) + f log ^| + C. 

Page 191, §101 

1. sin"* f^) + C. 2. | tan-i (~p) + C. 

3.ilo g J^f + C. 

#v /o __ 

4. -~ log [2a; + 1 + V±x 2 + 4x - 6] + C. 

7. ^ sin- *£» + C. 8. ^ log ^^ + C. 



9. 



^ log [2/ + 1+ V^ 2 + U + 2] + C. 



2 



10. log (a; 2 + 6a; + 25) - V tan" 1 ir~^\ + C 

11. 2V2z 2 + 2x - 3 + ^p log [(2a; + 1) +V4a; 2 + 4a; - 6] + C. 

12. 2 log (a; 2 + 2x + 5} + I- tan" 1 (^^) + C. 

13. - 2\/5 - 4a; - a; 2 + 5 sin" 1 (^p) + C. 

14. - fVSTSn^ - ^ sin" 1 (^) + C. 

. \/2 . _, / 4 - 3a; \ , „ 

15. - -2- sin '(-^J + C. 

1C V3, 3 (a; + 1) + V^a; 2 + 18a; + 9 , n 

16. -T^log- -— +C. 

17. sin" 1 (^p) + C. 

18. J log [2y + 3 + VV + 12y - 7] + C. 
19.^-log^ + C. 20.^ sin" 1 (^) + C. 



RESULTS OF EXERCISES 11 

21. - V2 + 3.r - z 2 + 4 sin" 1 ( 2x ~ 3 ^ + C. 

OQ x , 2 + 3z + V4 + 12z - 7x 2 „ 
23. — i log h C. 

Page 192, §102 

1. xfg (8a: 2 - 18s + 81) (2x + 3)* + C. 

2. |[Vx + 1 - 3] (x + 1)* + 4 tan" 1 (a + 1)* + C. 

3. 2VF+1 + -^log ^^ + I 3 -^ + C. 

V3 Vz + 3 + V3 

4. Wx - 2 + 3\/2 tan" 1 \^-y^ + C. 

5. 6 VT+1 - 4 log V + * — - + C. 

V a; + 1 + 1 

6. x + 1 + 4Vx + 1 + 4 log Wx + 1 - l) + C. 

7. 2 (6x* - Sx* + 2x&) - / f log (3x& + 1) + C. 

8. Mb* - 16) (x + 4)* +C. 

9. Hog ^L±^ 2 -f C. 

V a; + 4 + 2 

10. 3 V2T+~ 3 - 4 log y^l^ + C. 

V 3 V2x + 3 + V3 
11. ^[16x* + 15x*]a; + C. 



12. §\/2a7^3 + \,°V3 tan" 1 \^jp^ + <?. 

13. tVIGO/ 6 - 84/ 4 + 105J 3 + 140* 2 - I20t - 420]* + 3 log (t 2 + 1) 

+ 6 tan" 1 t + C, where t = (x + 2)* 

14. 2VaT^3 - 2\/7 tan" 1 yj x ~-- + C. 

15. ¥fr A - 3**] + 5 log (1 + x J ) + 10 tan" 1 x'° + C. 

16. W2T+3 + ^ log V3V2T+3 - Vg c 

9 VsV2x + 3 + Vl3 



12 RESULTS OF EXERCISES 

Page 194, §103, (a) 

1. — cos x + f cos 3 x — i cos 5 x + C. 

2. J sin 3 x — \ sin 5 x + C. 3. i sin 4 a; — J sin 6 x + C. . 

_3_ 7. 

4. f cos 3 # — cos x + C. 5. f (sin a;) 2 — f (sin x) 2 + C. 

6. + cos 7 a; - i cos 5 x + C, 7. 2 (sin a;) 2 - |(sin a;)* + C. 

8. - f(cos 0) 3 + i(cos 0)* - A (cos 0)^ + C. 

9. J sin 3 a. — \ sin 5 a + C. 

10. - | cos 3 (2x + 3) + j\ cos 5 (2a: + 3) + C. 

Page 195, §103, (b) 

1. i(2x - sin 2x) + C. 

2. A (24a; + 8 sin 4a: + sin 8a;) + C. 

3. T fa(12a; - 3 sin 4a: - 4 sin 3 2a;) + C. 

4. Jg (12a; - 8 sin 2a; + sin 4a;) + C. 

5. A (6a; - sin 6a;) + C. 

6. T k(60a; + 8 sin 10a: + sin 20a;) + C. 

Page 196, §103, (c) and (e) 

1. i sec 4 x — sec 2 x -\- log sec x + C. 

2. — i cot 3 x — cot x + C. 

3. | tan 3 x + | tan 5 a; + I tan 7 a; + C. 

4. | sec 7 a; — t sec 5 a; + C. 5. — J cot 3 a; + cot a; + x + C. 

6. — cot x — f cot 3 a; — i cot 5 a; + C. 

7. i tan 5 a: + C. 

8. tan a; + f tan 3 a; + \ tan 5 a; + C 

9. ^ sec 7 x — f sec 5 a; + J sec 3 a; + C 

10. \ tan 7 a + J tan 9 a; + C. 

11. | (tan a;) 3 + T 3 T(tan a?)"^" + C: 

12. | tan 3 a: + C. 

13. | (tan x)* + f(tan »)* + A (tan x)V + C. ■ 

14. |(sec 2 (9 - esc 2 (9) - log sin + log cos + 2(9 + C. 

15. tan - + C. 

16. i cot 3 19(3 tan 4 (9-6 tan 2 - 1) + C. 

Page 199, §104 

1. log (x + Va; 2 - a 2 ) + C. 



2. 1 ^ 2 da; = lo S (^ + A 


/a 2 + x\ 


3. AOa; 2 - 2) (1 + x*)* + C. 





RESULTS OF EXERCISES 13 

4. 1 log 1= + C. 

a a + V a 2 + x 2 

5. X log ?- + C. 

a a + V a 2 - a: 2 

6. -4= log 7 _ + C. 

V7 V7 + V7 - 3z 

jVx 2 + 9 

8. * + 1 + C. 9. , *~ — . + C. 

VI ~ z 2 V 4x - 3 - a; 2 

10. fa 2 [j sin 3 20 - i sin 4(9 + id] + C, where (9 = sin" 1 l^j 3 - 

11. ^[sin- 1 - - -Va 2 - x 2 ] + C. 

2 a a 2 

12. ^[9sin-i^ + V§*V9 - 5**] + C 

1U o 

13. ^[sec-i | + lV^^9] + C. 

14. V[3 sin-i| + ^(45 - 2x 2 )v / 9~^ 2 ] + C. 

16. A -7 -- +C. 

V 16 - x 2 

17. ^ — ^±JL= +C. 

Vi ! + 6x + 25 

18. j[27 sin-* ^~ - (x + 9) V6x - x 2 ] + C. 

19. - J ;. x+ = + c. 

Vz 2 + 4x - 5 

20. sin" 1 -^J + C. 

Vn 

Page 201, §105 

1 2 *3,r 9 7F . <* 9_ 4. 7F . 

1. T6-7T. *• 2 6 - **• 4 * O 

a/2 
5. 2 4 V. 6. >. 7. ^-f. 8. 0.0855. 

9 - w- 10 - i£ "■ **■ 12 - l {b2 - y2) - 



14 RESULTS OF EXERCISES 

Page 202, §106 
1. -q(3 log x — 1) -f C. 2. x sin x + cos x + C. 

3. re sin" 1 a; + Vl - x 2 + C. 4. 3Ve 4 *(8a; 2 - 4s + 1) + C. 

5. a; tan -1 a; — log Vl + a; 2 + C. 

6. sin a; — x cos a; + C. 

7. ~^ 2 [(r* + l)k>ga;-l]+C. 

8. ij [16a; 3 tan" 1 2a; - 4a; 2 + log (4a; 2 + 1)] + ft 

9. % sin a; (12 — sin 2 x) — \x cos a;(3 + cos 2 x) + C. 

10. a; (log a: - 1) + C. 

11. J (cos 2a; + 2a; sin 2a; - 2a; 2 cos 2x) + C. 

12. cos x{l — log cos x) + C. 

Page 203, §106 



8.602 



sin (7* - a) + C, where a = 125° 32'. 



2. ttvtt cos (8« - a) + C, where a = 110° 34'. 
8.544 ' 

3. ^j sin (3* -a)+C } where a = 99° 28'. 

4. -f^ cos (4* - a) + C, where a = 92° 50'. 
4.UU0 

e~ x 

5. — -=. sin (a; — a) + C, where a = 135°. 

V2 

6. -r^r cos (5* - a) + C, where a = 121°. 

g-0.4* 

7. — ; sin (cot — a) + C. 

Vco 2 +0.16 

e -0 .2* 

8. — -. COS (a>£ — a) + C. 

V" 2 + 0.04 

9. Y^T cos (5* - a) + C, where a = 94° 35'. 

10. |^- sin (U -a)+C, where a = 94° 17'. 

Page 204, §108 

1. — % [esc x cot x + log (esc a; + cot x)\ + C. 

2. i sec 3 a; tan x + Msec ai tan a; + log (sec a; + tan a;)] + C. 



J. a 4 I sec 5 a 



da + C, where a = sec -1 -. 
a 



RESULTS OF EXERCISES 15 



4 



a 2 I sec 3 a da. 
Jo 



5. a 2 I sec 3 a da + C, where a = tan x L . 

J 

/# — 2 
sec 3 a da + C, where a = tan -1 — j=— 
V7 

7. a 4 I sec a da. 

Jo 

8. 9 I (sec 3 a — sec a)da. 

Jo 

Page 208, §109 
1. tV 2. 5«ftir. 3. Hf. 4. Att. 5. sV. 

6. .¥^. 7. Iff. 8. if. 9. rf 5. 10. A- 

11. ~. 12. £. 13. sfer. 14. rf*. 

15. T 3 67ra 4 . 16. J-Tra 6 . 17. ^wa\ 18. T f 5 a 3 . 

19. Iwa 2 . 20. Jtt. 21. fTra 4 . 22. WaK 

Page 211, §110 

2+tan ^ / x\ 

1. i log + C. 2. § tan" 1 (2 tan ^) + C. 

o _ + _ x \ 2/ 



2 - tan ■ 

tan - — 2 
3. J log +C. 

2 tan | - 1 

1 -^tai —7— 

6. - i log (1 - 3sini) + C. 

tan * + 2 - V3 

7. 2x + * log - - + C. 

V3 tan? +2 + a/3 



f2 tan % + 1~| 

- — Jr- - t 



,-, fe] + c. 



16 RESULTS OF EXERCISES 

8. i esc 2 x + tan -= — | sec x cot x — § log (esc # + cot #) + C. 

io. |io g ^ ana; 7} + c. 

3 tan £ + 1 

Page 217, §111 

1. tV log (x - 2) + A log (x 2 + 2a: +5) + tt tan"* (^^) + & 

2. log (a + l) 2 + g±± + C. 

3. _ # fog * + f log (x - 3) + J ^^ + C. 
4 v ^log|^ 2 + ftan-i| + a 

5. f log (* - 1) - f log (x 2 - 2x + 5) + tan" 1 (^^) + C. 

6. flog (x + 4) + ft log (2x + 1) - I l + C. 

{Zx — 1) 

7. log Vx 2 + 1 + tan" 1 s - 3(z 2 + 1) _1 + C. 

8. 1(2 + l) 2 - | log x + V- log (a + 3) + V log (x - 3) + C. 

Page 219, §112 
1. lira 2 . 2. a 2 . 3. §7ra. 

4. 10rr. 5. iwa\ 6. 49tf. 

7. 47ra 2 . 8. 200 foot-pounds. 9. 300 pounds. 

10. 6a. 12. Iwab. 13. Ixa 2 . 

14. 8a. 15. &ra 2 b. 16. 11.08. 



17. 


\ira 2 sinh — - + 
a 


wax. 








18. 


37ra 2 . 


19. 


2.078. 


20. 


\ira z sinh 


2b 
a 


+ i7ra 2 6. 




21. 


3h*a*. 


22. 


6f. 






23. 


72. 


24. 


2. 


25. 

-1]. 


^i 72 
Clog^. 






27. 
29. 


3tt. 


28. 


^ 1 + V- 

a 


Vl +a 2 
a 


30. 


#7ra 2 . 


31. 


25tt. 






32. 


fTra 2 . 


33. 


4tt - 21|. 


34. 


9704 foot-pounds. 


35. 


9498. 


36. 


63,540 pounds. 


37. 


a 2 sinh- 1. 











38. £ [2tt Vl + 4tt 2 + log (2tt + Vl + 4tt 2 ). 



39, 



RESULTS OF EXERCISES 17 

a I y/l — e sin 2 u du, where e is the eccentricity of the ellipse, 
Jo 





u 


= - — d, and a > b 


40. (512) -(2340.4)*-. 


41. ottW. 




42. ,Va 3 (3ir - 4). 


43. iwkr 2 h 2 . 




44. 4690 foot-pounds. 


45. ¥*•. 




46. a sinh — . 
a 


47. 8a. 




48. 27r 2 a 2 6. 


49. 3-f T a 2 (207r 


+ 1 - 39V3). 



50. 184,300 foot-pounds. 51. -^a 3 . 

52. y-rra 2 . 53. llx. 54. 2.278*- - 3\/o. 

55. -V 6 a 2 . 56. hraK 

Page 226, §113 
1. 2. 2. No limit. 3. No limit. 4. 2 a/2. 

5. No limit. 6. 2 y/a. 7. Jtt. 8. §x. 

9. No limit. 10. 6. 11. 3(2* + 3 f ). 12. fxa 2 . 

Page 227, §114 
1. 1. 2. 1. 3. 1. 4. 2. 5. 47ra 2 . 

Page 230, §116 
1. a/29. 2. V40. 3. Vl3. 4. Vl3. 5. Vl30. 6. Vl4. 

Page 231, §117 

1. 1V3, iV3, |V3. 2. -^, ^|, -^. 

o - 1 2 - 3 n 

3 - 77^> TTtv 77?7- 4. jV3, jV3, *V3. 



5. 



Vl4' Vl4' Vl4' 

- r - 7 4 

V66' V66' a/66' 



Page 232, §118 



Q — 90 

1 and 2. -^=. 2 and 3. —7=. 

V87 V406 

2 



3 and 4. 



V42* 



Page 233, §119 

1. x — -y/2y —2 = 4, if the normal extends into the eighth octant. 

2. x - y + V22 + 10 = 0. 



18 RESULTS OF EXERCISES 

Page 234, §120 

1. — 7=z 7=2/ - —7= z = —7=; — ■ f=; eighth. 

V14 V14 V14 V14 V14 

2. - §V3z - Wly - Wh = WS; iVS] seventh. 

. 1 3,2 33",, 

3. —7= a 7= y H —=_ z = —y=; -7=; fourth. 

V14 V14 V14 Vl4 Vl4 

1 *? R ft 

6. — 7= x -\ 7= 2/ = — 7= J — 7^ J in the :n/-plane between the first and 

V5 V5 V5 V5 

fifth octants. 



Page 236, §121 

- 1 " q ' q ^ q *•• 7 ~ _ 7 ' 7 a. 

*5 o *> 2^—3-4 



Page 326, §122 
1. 79° 7'. 2. 122° 19'. 

Page 238, §124 
1. 4VI4. 2. |V6. 3. HV38. 

Page 240, §125 
z-f y — # z - 



2. 



V26 V26 V26 

z-f_2/-0_ g - | 



V26 V26 a/26 
x - _ y - _ z - 



1 



V35 V35 V35 

Page 241, §126 



2. ?/2 + g 2 = (3.2 _ a 2)2 # 

Page 249, §130 



1. (a) 0; ?f ; - - 2 - 

2/ 3 2/ 2 

2. (a) 2s. (6) — : L 



RESULTS OF EXERCISES 19 

Page 252, §132 



1. 


3. 


2. i 3. - ^ 4. Irf. 


5. 


ia 2 (w - 2). 


6. 15,750 7. fa 3 . 8. fa 1 - |a« 


9. 


_6_ 
35' 


10. i 11. A- 12. |7ra 3 . 
Page 255, §133 


1. 


•tV 


3. Aa 2 (3* - 8). 4. T Va 2 (37r - 8) 


6. 


3i 


8. 0.879. 

Page 258, §134 


1. 


2oV / 27r. 


2. i 3. V-a 3 - 4. T V. 
Page 260, §135 


1. 


9 T 


2. 37ra 2 . 3. a 2 . 4. wa 2 . 
Page 262, §136 


1. 


¥^ 3 . 


2. \mr z . 3. |aoc. 


4. 


ra /•(«§_ 


'' dz dy dx. 




Jo Jo 


Jo 


5. 


87ra 3 . 


6. }f. 7. iV27r. 
Page 270, §140 


1. 


£ = y = A 




2. 


Part Li = 


io « -■ s p + o ~ — 4a ~ _ a 


21 ,y 12 . fart 3. x 5(3?r _ 8 y 2/ 3?r _ g 




Part 6. x = 


s f£, £ = tV 


4. 


- 4r 

07T 


7. x = y = — 11. x = Jr. 

7T 


12. 


X = fk 


13. 5 = fa. 


14. 


- (6/i - 
X = 


3 

a) (4/i + a)'- + a 2 



10 [(4fc +a)- - a-] 

15. On a line joining the apex with the centroid of the boundary of 
the base, at a point two-thirds of the way from the apex to the 
centroid. 

16. x = -n-a, y = . '/. 17. x = ira, y = in. 
io - - 256a - 

18 - x = 1J = 31&T 19 - £ = IJ = '"■ 

23. z = y /t7to, x = fa. 



20 RESULTS OF EXERCISES 

Page 275, §141 

— — An 
1. 47r 2 a&. 2. 2ir 2 a 2 b. 4. x = y = J?" 

Sir 

Page 276, §142 

1. Part 1. x = %a } y — 0. Part 2. x = %a, y = -^ — 

rt - A 76 v 2a rt 2a sin a 

2. x = f a 777^ 3. 



4. x = y = | 



105 7T 3a 

a 



Page 282, §145 

1. \ab(a 2 + 6 2 ); T Va6(a 2 + b 2 )) £a& 3 ; ^afc 3 . 

2. T Va6 3 ; 3Va6 3 . 3. #7ra 4 . 4. ixa 4 . 5. §7ra 4 . 

6. A. 7. i P L 3 ; t VpL 3 . 8. ifcM 10. JraL 2 sin 2 a. 

11. 2ar 3 ; fr 3 (2a - sin 2a); §r 3 (2a + sin 2a). 

12. \bh\ 13. ^bh\ 
14. i7ra6 3 ; i7ra 3 &; i*-a&(a 2 + ¥). 

Page 283, §146 

1 35^4. 2 Va 4 O a 4 (457T - 128) 

1. T6 -7ra , 3^a . 2. 192Q 

3. |a 4 (2a - sin 2a). 4. Ja 3 (2a - sin 2a). 
5. |aa 4 ; \ol& — % 

Page 287, §147 

irb 2 h{Ab 2 + h 9 ) , irb 2 h{W + 2fr 2 ) 
80 ; 60 

2. =§ (3r 2 + 4/* 2 ); ^ (3r 2 + &*>. 

3. j^abcic 2 + & 2 ). 4. ±7rr 4 /*. 

5. MR 4 ~ r*)h; ^ (R 2 - r 2 ) (SR 2 + 3r 2 + Ah 2 ); 

j| (£ 2 - r 2 ) (3# 2 + 3r 2 + A 2 ). 

11. ^irr% 12. A^^(3a 2 +L 2 ). 

14. iV3a 4 ; |\/3a 4 . 
Page 292, §151 

2. _ 6 3 . 

V#[4 + 9a;] I 



8. 


t 8 5tt(# 5 - r 5 ). 


13. 


T V7ra 4 . 


1. 


2 


[5 -&c + 4z 2 ]f 



RESULTS OF EXERCISES 21 

3. 2x% 4, 2(1 ~ 3x) 



{x* + 1]J [1 + (2s - 3s 2 ) 2 ]! 

. 18s° ■> 

0. , D. 



[x« + 36]^ [4s + l]i 

7 2x| 30z 4 



9. 



3[xi + 1]! " [4z 5 + 9]^ 

6s 2 
[4z 3 + l]t 

Page 293, §152 



1 o 1 t 

• 2* — ■ CSC — ' 

" a[sinh 2 1 + cosh 2 t]i * 4a ' 2 

Page 296, §152 

1. 216x 2 = (Sy - 3) 3 . 

Page 299, §155 

1. y 2 = 4px. 2. x 2 + y 2 = a 2 . 

3. x 2 — 4ty 2 = 0, two straight lines. 

2 2 2 

4. x 3 + y 3 = a 3 j where a is the constant length and the coordinate 
axes are the mutually perpendicular lines. 

5. x 2 = 4?/. 6. y 2 = 4p(x + p). 

Page 313, §163 

/>»2 />«4 /*»6 

l.coss = l- | - I -+ | - I - | - I + . . . 

(# — a )2 

2. cos x = cos a — (x — a) sin a — - — r-= — cos a 

'_ 

, (x - a) 3 . , 

H r^ — sin a + . . . 

h 2 h z 

3. cos (a + h) — cos a — h sin a — ,-r- cos + 1-5- sin a + . . . 

I ^ [_£_ 

4. sin (a + h) = sin a + h cos a — nr sin a — r«- cos a + . . . 

«_ LA 

5. Fart 1. — . cos x 6 ; Part 2. — . sin z 3 ; 

/i 3 /i 3 

Part 3. . _- sin :r 3 ; Part 4. — . cos x 3 . 

x 2 . x 3 . x 4 



,.«..! + , + -+-+£.+ 



22 



RESULTS OF EXERCISES 



e a 



e a 



1 + (x - a) + 



7. e 

8. 6 

9. log (1 + x) — x - 
10. log (1 - x) = - \ 






(a; — a) 2 (x — a) 



12 



+ 



13. 



+ 



]• 



a^ a) d 

2" + "3 ~ ' * ' 

* + T + T + 



]• 



11. tan -1 a? = x 



x 6 x* 

3 + 5 



+ 



12. 
13. 

1. 

4. 

7. 
10. 
12. 
14. 



(a) 1.39561; (b) 1.105171; (c) 0.052336; (d) 0.5403. 



0.8387. 

Convergent. 
Divergent. 
Convergent. 
For all values. 
For all values. 
- 1 < x < + 1. 



- 2 + 2i. 
1. 



14. 0.5299. 
Page 319, §165 
2. Divergent. 3. Divergent. 

5. Convergent. 6. Convergent. 

8. Convergent. 9. Convergent. 

11. For all values. 

13. - 1 < x < + 1. 



2. -* + 



Page 323, §169 

^* 

2 '* 

5. 1. 

Page 324, §169 



3. - 1. 
6. - 125. 



9-n-i 



2. V2e8, and V2e 8 



1. 

5. 

9. 
13. 
17. 
21. 
25. 

12. 
13. 
14. 
15. 



1. 

n. 



co ? if n > 0. 



1. 



Page 327, F §170 

2. i 3. 0. 

6. f. 7. 3. 

10. f. 11. 0. 

14. 0. 15. oo. 

18. 1. 19. J. 

22. 1. 23. 1. 



4. 3. 

8. - 6. 

12. 0. 

16. oo. 

20. 1. 

24. 1. 



Page 331, §171 

372 cubic inches per hour. 

— 8.4 pounds per square foot. 

— 25.4 pounds per square foot per second. 
5.32 cubic feet per second. 



RESULTS OF EXERCISES 23 

Page 336, §173 
1. x 3 y 2 = C. 2. sin - = C. 3. Not exact. 

y 

4. e x » = C. 5. Not exact. G. —„ + x - y 2 = C. 



yc 



7. Not exact. 



Page 340, §178 
2. cos y = C cos x. 

3. [x + vtt^] [2/ + vTT7 2 ] = c vtt^. 

4. 5x 2 y = Cy -2. 

5. a; + y + 4 = C(x2/ + 2a; + 2 27 + 3). 



7. log (z?,) = x — y + C. 


8 -'-V 


= e x + C. 


9. log (xy) + \ - 1 ■ = C. 
x y 


10. log {x 2 y) 


= y + C. 



11. sin 2/ = C(l - e*) 3 . 

Page 242, §179 

1. log y = ^ + C. 2. 32/ 3 log 2/ = x 3 + C?/ 3 



x 



a; 



3. (y + 2x) 3 (y + x) 2 = C. 4. log y + ^ = C. 

5. y + Va^TT 2 = ^ 2# 6# log x + sin 1 = c 

7. sin" 1 V r = log x + C. 8. log 2/ + - = C. 

x 2/ 

Page 344, §180 

1. ye x2 f= x + C. 2. 2/ = 2(sin a; - 1) + Ce~ 6in x . 

3. ?/ = tan x - 1 + Ce" tan *. 4. 3?/(l + a, 2 ) = 4x 3 + C. 

5. 62/(1 + x) 2 = (1 + z) 6 + C. 6. y - az + CVl - x 2 . 

1 

7. ?y = x"(e* + C). 9. 2/ = Cx 2 e* -f a; 2 . 

Page 345, §181 

1. sr« = |s 3 + Cx 5 . 2. y~ 2 = x + J + Ce 2x . 

3. 49?/ 3 T 7(x + 1) + 1 = Ce lx . 4. t/" 1 + a = CVl - x 2 . 

6. y~ * + ?z 3 = Cxi 6. I/" 1 = 1 + log x + Cx. 

7. y*(x + l) 6 = ^x 10 + 2x 9 + 4 8 ^x 8 + 6 7 °^ 7 + l i~x* + V* 5 + i* 4 + C. 



24 



RESULTS OF EXERCISES 



Page 349, §183 
1. y = Cl e" + c 2 e->*. 2 . y = C x #* + C 2 e~*. 

3. y = C l6 * + C*'*. 4 . j, - C l6 ». + C 2 e~^ 

5. 2/ = Cie 7 * + C 2 . 



I. V = e 2 *(Ci + (7 2 z). 
3. 2/ = Ci + C 2 z. 



1. 2/ = •"*■ [ 



Page 350, §184 

2. y = e 4 *(d + Caa?). 

Page 351, §185 



A cos 



— 2~ a; + # sm — — - £ J 



2. ?/ = A cos 3z + £ sin 3*. 
4. = A cos o>2 + .6 sin ut. 



3. 2/ = A cos 3 + £ sin x. 



/-~ 



LIBRARY OF CONGRESS 

HIM Hill Hill Hill lllll llll Hill Hill lllll Hill lllll llll llll 




003 527 321 9 * 



LIBRARY OF CONGRESS 



003 527 321 9